Can You Use Trig On A Non Right Triangle

The world of trigonometry extends far beyond the confines of right triangles, offering a powerful toolkit for analyzing any triangle, regardless of its angles. While the basic trigonometric ratios—sine, cosine, and tangent—are defined within the context of right triangles, more advanced concepts like the Law of Sines and the Law of Cosines allow us to solve for unknown sides and angles in any triangle, opening up a vast range of applications in fields like surveying, navigation, physics, and engineering.

The Limitations of Basic Trigonometry with Non-Right Triangles

Traditional trigonometric ratios (SOH CAH TOA) are based on the relationships between the sides and angles within a right triangle.

• Sine (sin θ) = Opposite / Hypotenuse
• Cosine (cos θ) = Adjacent / Hypotenuse
• Tangent (tan θ) = Opposite / Adjacent

These definitions rely on the presence of a 90-degree angle (the right angle) and the identification of the hypotenuse (the side opposite the right angle). In a non-right (oblique) triangle, there's no right angle or hypotenuse, making these basic ratios directly inapplicable. Trying to force these ratios onto non-right triangles would lead to incorrect results and a misunderstanding of the triangle's geometry. Therefore, alternative methods are needed.

Introducing the Law of Sines

The Law of Sines establishes a relationship between the sides of a triangle and the sines of their opposite angles. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles in a triangle.

The Formula:

a / sin(A) = b / sin(B) = c / sin(C)

Where:

• a, b, and c are the lengths of the sides of the triangle.
• A, B, and C are the angles opposite to sides a, b, and c, respectively.

When to Use the Law of Sines:

The Law of Sines is particularly useful when you know:

• Two angles and one side (AAS or ASA): Knowing two angles automatically gives you the third (since the angles of a triangle sum to 180°). With one side known, you can use the Law of Sines to find the other two sides.
• Two sides and an angle opposite one of them (SSA): This is known as the ambiguous case because there might be zero, one, or two possible triangles that satisfy the given conditions. Careful analysis is required to determine the correct solution(s).

Example 1: Solving for a Side (AAS)

Suppose you have a triangle where:

• Angle A = 30°
• Angle B = 70°
• Side a = 8 units

You want to find the length of side b.

1. Calculate Angle C: C = 180° - A - B = 180° - 30° - 70° = 80°
2. Apply the Law of Sines:

   a / sin(A) = b / sin(B)
   8 / sin(30°) = b / sin(70°)
   

3. Solve for b:

   b = (8 * sin(70°)) / sin(30°)
   b ≈ (8 * 0.9397) / 0.5
   b ≈ 15.035 units
   

Example 2: Solving for an Angle (ASA)

Suppose you have a triangle where:

• Angle A = 45°
• Side b = 12 units
• Angle C = 60°

You want to find the measure of angle B.

1. Calculate Angle B: B = 180° - A - C = 180° - 45° - 60° = 75°
2. Apply the Law of Sines: (While you could solve for a side first, since you want angle B, and you know side b, you can actually skip directly to this step if you wish.)

   b / sin(B) = c / sin(C)  //(We could also use a/sin A, but don't know 'a' yet.)
   12 / sin(75°) = c / sin(60°)
   

3. (Optional) Solve for c: (Though not necessary to find angle B):

   c = (12 * sin(60°)) / sin(75°)
   c ≈ (12 * 0.866) / 0.9659
   c ≈ 10.77 units
   

The Ambiguous Case (SSA):

The SSA case is tricky. Given two sides and an angle opposite one of them, there might be:

• No solution: The side opposite the given angle is too short to reach the third side.
• One solution: The side opposite the given angle is just long enough to form a single triangle, or it's longer than the adjacent side.
• Two solutions: The side opposite the given angle is long enough to intersect the third side in two places, forming two distinct triangles.

To determine the number of solutions in the SSA case, you typically need to calculate the height (h) of the triangle from the given angle to the unknown side. Then compare the length of the side opposite the angle (let's call it 'a') to the height (h) and the length of the adjacent side (let's call it 'b'):

• If a < h: No solution.
• If a = h: One solution (a right triangle).
• If h < a < b: Two solutions.
• If a ≥ b: One solution.

Example 3: The Ambiguous Case

Suppose you have a triangle where:

• Angle A = 30°
• Side a = 6 units
• Side b = 10 units

1. Calculate the Height (h): h = b * sin(A) = 10 * sin(30°) = 10 * 0.5 = 5 units
2. Compare 'a' to 'h' and 'b': Since h < a < b (5 < 6 < 10), there are two possible solutions.

To find both solutions, you'd proceed with the Law of Sines to find angle B, remembering that sin(B) has two possible angles between 0° and 180° that yield the same sine value (one acute and one obtuse). You'd then calculate the remaining angles and sides for each possible triangle.

Introducing the Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It's a generalization of the Pythagorean theorem and is essential when you cannot use the Law of Sines.

The Formula:

There are three forms of the Law of Cosines, each focusing on a different angle:

• a² = b² + c² - 2bc * cos(A)
• b² = a² + c² - 2ac * cos(B)
• c² = a² + b² - 2ab * cos(C)

Where:

• a, b, and c are the lengths of the sides of the triangle.
• A, B, and C are the angles opposite to sides a, b, and c, respectively.

When to Use the Law of Cosines:

The Law of Cosines is particularly useful when you know:

• Three sides (SSS): You can use the Law of Cosines to find any of the three angles.
• Two sides and the included angle (SAS): You can use the Law of Cosines to find the third side. Once you have all three sides, you can use the Law of Sines (or Law of Cosines again) to find the remaining angles.

Example 4: Solving for a Side (SAS)

Suppose you have a triangle where:

• Side a = 5 units
• Side b = 8 units
• Angle C = 77°

You want to find the length of side c.

1. Apply the Law of Cosines:

   c² = a² + b² - 2ab * cos(C)
   c² = 5² + 8² - 2 * 5 * 8 * cos(77°)
   

2. Solve for c:

   c² = 25 + 64 - 80 * cos(77°)
   c² ≈ 89 - 80 * 0.225
   c² ≈ 89 - 18
   c² ≈ 71
   c ≈ √71
   c ≈ 8.43 units
   

Example 5: Solving for an Angle (SSS)

Suppose you have a triangle where:

• Side a = 7 units
• Side b = 9 units
• Side c = 12 units

You want to find the measure of angle A.

1. Apply the Law of Cosines (rearranged to solve for cos(A)):

   a² = b² + c² - 2bc * cos(A)
   cos(A) = (b² + c² - a²) / (2bc)
   cos(A) = (9² + 12² - 7²) / (2 * 9 * 12)
   

2. Solve for cos(A):

   cos(A) = (81 + 144 - 49) / 216
   cos(A) = 176 / 216
   cos(A) ≈ 0.8148
   

3. Solve for A:

   A = arccos(0.8148)
   A ≈ 35.44°
   

Choosing Between the Law of Sines and the Law of Cosines

• Law of Sines: Use when you have an angle and its opposite side (or can easily find one). Good for AAS, ASA, and sometimes SSA (but be careful of the ambiguous case!).
• Law of Cosines: Use when you don't have an angle and its opposite side. Essential for SSS and SAS.

In some cases, you can use either law. For example, after using the Law of Cosines to find a side in an SAS triangle, you could then use the Law of Sines to find one of the remaining angles.

Beyond the Basics: Applications

The Law of Sines and the Law of Cosines have numerous real-world applications:

• Surveying: Calculating distances and angles in land surveys, even when direct measurement is impossible.
• Navigation: Determining the position and heading of ships, aircraft, and other vehicles.
• Engineering: Designing bridges, buildings, and other structures, ensuring stability and accuracy.
• Physics: Analyzing forces, vectors, and motion in various physical systems.
• Astronomy: Calculating distances to stars and planets.

Example: Surveying

A surveyor needs to determine the distance across a river. She stands at point A on one side of the river and sights a point C on the opposite bank. She then walks 100 meters along the riverbank to point B and measures the angle ABC to be 60° and the angle BAC to be 40°. How wide is the river (i.e., the distance from A to C)?

1. Calculate Angle ACB: ACB = 180° - 60° - 40° = 80°
2. Apply the Law of Sines:

   AB / sin(ACB) = AC / sin(ABC)
   100 / sin(80°) = AC / sin(60°)
   

3. Solve for AC:

   AC = (100 * sin(60°)) / sin(80°)
   AC ≈ (100 * 0.866) / 0.9848
   AC ≈ 88 meters
   

The river is approximately 88 meters wide.

Other Trigonometric Tools for Non-Right Triangles

While the Law of Sines and Law of Cosines are the most common tools, other trigonometric relationships can be used for solving non-right triangles:

• Area Formulas: Several formulas exist to calculate the area of a triangle when you don't know the height directly:

  • Area = (1/2) * ab * sin(C) (where a and b are two sides and C is the included angle). This is extremely useful in SAS triangles.
  • Heron's Formula: Area = √(s(s-a)(s-b)(s-c)) where s is the semi-perimeter (s = (a+b+c)/2). This is useful in SSS triangles.

• Projection Theorem: This relates the sides of a triangle to the projections of one side onto the other two sides.

• Tangent Rule: This involves the tangents of the half-angles of a triangle and the lengths of the sides.

Conclusion

While basic trigonometric ratios are limited to right triangles, the Law of Sines and the Law of Cosines extend the power of trigonometry to all triangles. These laws, along with area formulas and other trigonometric relationships, provide a comprehensive toolkit for solving for unknown sides, angles, and areas in any triangle, making them indispensable tools in various fields that rely on geometric analysis and spatial reasoning. Understanding when and how to apply these laws is crucial for solving a wide range of practical problems. Always remember to consider the ambiguous case when using the Law of Sines in the SSA scenario and to carefully choose the appropriate law based on the given information. Mastery of these concepts opens up a world of possibilities in fields ranging from surveying and navigation to physics and engineering.